\documentclass[11pt]{report}
\usepackage{amsmath,amsfonts,amssymb,bm}
\usepackage{graphicx}
\usepackage{algorithm, algorithmic}

\renewcommand*\thesection{\arabic{section}}
\newcommand{\EqnArr}[1]{\begin{align}#1\end{align}}
%\newcommand{\Prob}{\mathrm{Pr}}
\newcommand{\Cal}[1]{\mathcal{#1}}
\newcommand{\Rm}[1]{\mathrm{#1}}
\newcommand{\Bb}[1]{\mathbb{#1}}
\newcommand{\Br}[1]{\left(#1\right)}
\newcommand{\ClBr}[1]{\left\{#1\right\}}
\newcommand{\SqBr}[1]{\left[#1\right]}
\newcommand{\Norm}[1]{\left\|#1\right\|}
\newcommand{\Trm}[1]{\textrm{#1}}
\newcommand{\Prob}[1]{\mathrm{Pr}\ClBr{#1}}
\renewcommand{\Bar}[1]{\overline{#1}}
\newcommand{\xx}{\bm{x}}
\newcommand{\qq}{\bm{q}}
\newcommand{\QQ}{\bm{Q}}
\newcommand{\partf}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\What}[1]{\widehat{#1}}

\newtheorem{thm}{Theorem}
\newtheorem{defi}{Definition}

\title{Useful formula and theorems}
\author{Long Q Tran}

\begin{document}
\maketitle

\begin{thm}[Neother's theorem] If the action
\EqnArr{
A = \int L(\qq, \dot{\qq}, t) dt
}
is invariant under transformations (up to first order)
\EqnArr{
t &\leftarrow t + \sum_r \epsilon_r T_r\\
\qq &\leftarrow \qq + \sum_r \epsilon_r \QQ_r,
}
where $r = 1,2,\ldots,N$, then the quantities
\EqnArr{
\Br{\partf{L}{\dot{\qq}}\cdot\dot{\qq} - L}T_r - \partf{L}{\dot{\qq}}\cdot\QQ_r
}
are conserved.
\end{thm}

\begin{defi}[von Mises-Fisher distribution] The density function is
\EqnArr{
f_p(\xx; \bm{\mu},\kappa) = \delta(\|\xx\|_2-1) C_p(\kappa) 
\exp( \kappa \bm{\mu}\cdot \xx )
}
with $\kappa\geq 0, \|\bm{\mu}\|_2 = 1$. The normilizing factor is
\EqnArr{
C_p(\kappa) &= \frac{\kappa^{p/2-1}}{(2\pi)^{p/2}I_{p/2-1}(\kappa)}\\
C_3(\kappa) & = \frac{\kappa}{4\pi \sinh \kappa}.
}
The MLE is
\EqnArr{
\What{\bm{\mu}}&=\frac{\sum_n \xx_n}{\Norm{\sum_n \xx_n}}\\
\What{\kappa} &= \frac{\overline{R} (p-\overline{R}^2}{1-\overline{R}^2}
}
where $\overline{R} = \Norm{\sum_n \xx_n}/N$.
\end{defi}

%\begin{figure}[htb!]
%\centering
%\includegraphics[width=\textwidth]{../fig/markov_diff.eps}
%\caption{Difference to ground truth over iterations (markov chains)}
%\label{fig:Hidden Markov Model markov diff}
%\end{figure}

\end{document}